On Topological Entropy of Piecewise Smooth Vector Fields 2

TL;DR

This paper introduces a new framework for analyzing chaos in non-smooth vector fields by defining topological entropy, enabling the study of complex behaviors like chaos that are absent in smooth systems.

Contribution

It develops a novel approach to measure chaos in non-smooth vector fields through topological entropy, including constructing a trajectory space and deriving general results.

Findings

Established a metric space of trajectories for non-smooth vector fields

Defined topological entropy in this new setting

Provided examples with positive finite and infinite entropy

Abstract

Non-smooth vector fields does not have necessarily the property of uniqueness of solution passing through a point and this is responsible to enrich the behavior of the system. Even on the plane non-smooth vector fields can be chaotic, a feature impossible for the smooth or continuous case. We propose a new approach towards a better understanding of chaos for non-smooth vector fields and this is done by studying the entropy of the system. In this work we set the ground for one to begin the study of entropy for non-smooth vector fields. We construct a metric space of all possible trajectories of a non-smooth vector field, where we define a flow inherited by the vector field and then define the topological entropy in this scenario. As a consequence, we are able to obtain some general results of this theory and give some examples of planar non-smooth vector fields with positive (finite and infinite) entropy.